This document provides the table of contents for a textbook on advanced macroeconomics. The textbook is organized into 12 chapters covering various macroeconomic models and topics, including economic growth, business cycles, consumption, investment, unemployment, inflation, and fiscal policy. It also includes an epilogue on the 2008 financial crisis. Several chapters include empirical applications sections that discuss real-world evidence related to the theoretical models presented.
4. The McGraw-Hill Series in Economics
ESSENTIALS OF
ECONOMICS
Brue, McConnell, and Flynn
Essentials of Economics
Second Edition
Mandel
Economics: The Basics
First Edition
Schiller
Essentials of Economics
Eighth Edition
PRINCIPLES OF ECONOMICS
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Economics,
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Managerial Economics
Tenth Edition
INTERMEDIATE
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Microeconomics
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Macroeconomics
Eleventh Edition
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Microeconomics and
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ADVANCED ECONOMICS
Romer
Advanced Macroeconomics
Fourth Edition
MONEY AND BANKING
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Money, Banking, and
Financial Markets
Third Edition
URBAN ECONOMICS
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Urban Economics
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An Introduction
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International Economics
Seventh Edition
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A Reader
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6. ADVANCED MACROECONOMICS, FOURTH EDITION
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Library of Congress Cataloging-in-Publication Data
Romer, David.
Advanced macroeconomics / David Romer. — 4th ed.
p. cm.
ISBN 978-0-07-351137-5
1. Macroeconomics. I. Title.
HB172.5.R66 2012
339—dc22
2010040893
www.mhhe.com
9. ABOUT THE AUTHOR
David Romer is the Royer Professor in Political Economy at the Univer-
sity of California, Berkeley, where he has been on the faculty since 1988.
He is also co-director of the program in Monetary Economics at the National
Bureau of Economic Research. He received his A.B. from Princeton Univer-
sity and his Ph.D. from the Massachusetts Institute of Technology. He has
been on the faculty at Princeton and has been a visiting faculty member
at M.I.T. and Stanford University. At Berkeley, he is a three-time recipient
of the Graduate Economic Association’s distinguished teaching and advis-
ing awards. He is a fellow of the American Academy of Arts and Sciences,
a former member of the Executive Committee of the American Economic
Association, and co-editor of the Brookings Papers on Economic Activity.
Most of his recent research focuses on monetary and fiscal policy; this work
considers both the effects of policy on the economy and the determinants
of policy. His other research interests include the foundations of price stick-
iness, empirical evidence on economic growth, and asset-price volatility. He
is married to Christina Romer, with whom he frequently collaborates. They
have three children, Katherine, Paul, and Matthew.
13. CONTENTS
Preface to the Fourth Edition xix
Introduction 1
Chapter 1 THE SOLOW GROWTH MODEL 6
1.1 Some Basic Facts about Economic Growth 6
1.2 Assumptions 10
1.3 The Dynamics of the Model 15
1.4 The Impact of a Change in the Saving Rate 18
1.5 Quantitative Implications 23
1.6 The Solow Model and the Central Questions of
Growth Theory 27
1.7 Empirical Applications 30
1.8 The Environment and Economic Growth 37
Problems 45
Chapter 2 INFINITE-HORIZON AND
OVERLAPPING-GENERATIONS
MODELS 49
Part A THE RAMSEY–CASS–KOOPMANS MODEL 49
2.1 Assumptions 49
2.2 The Behavior of Households and Firms 51
2.3 The Dynamics of the Economy 57
2.4 Welfare 63
2.5 The Balanced Growth Path 64
2.6 The Effects of a Fall in the Discount Rate 66
2.7 The Effects of Government Purchases 71
Part B THE DIAMOND MODEL 77
2.8 Assumptions 77
2.9 Household Behavior 78
2.10 The Dynamics of the Economy 81
xi
14. xii CONTENTS
2.11 The Possibility of Dynamic Inefficiency 88
2.12 Government in the Diamond Model 92
Problems 93
Chapter 3 ENDOGENOUS GROWTH 101
3.1 Framework and Assumptions 102
3.2 The Model without Capital 104
3.3 The General Case 111
3.4 The Nature of Knowledge and the Determinants of
the Allocation of Resources to RD 116
3.5 The Romer Model 123
3.6 Empirical Application: Time-Series Tests of
Endogenous Growth Models 134
3.7 Empirical Application: Population Growth and
Technological Change since 1 Million B.C. 138
3.8 Models of Knowledge Accumulation and the
Central Questions of Growth Theory 143
Problems 145
Chapter 4 CROSS-COUNTRY INCOME
DIFFERENCES 150
4.1 Extending the Solow Model to Include Human
Capital 151
4.2 Empirical Application: Accounting for Cross-Country
Income Differences 156
4.3 Social Infrastructure 162
4.4 Empirical Application: Social Infrastructure and
Cross-Country Income Differences 164
4.5 Beyond Social Infrastructure 169
4.6 Differences in Growth Rates 178
Problems 183
Chapter 5 REAL-BUSINESS-CYCLE THEORY 189
5.1 Introduction: Some Facts about Economic
Fluctuations 189
5.2 An Overview of Business-Cycle Research 193
5.3 A Baseline Real-Business-Cycle Model 195
5.4 Household Behavior 197
5.5 A Special Case of the Model 201
5.6 Solving the Model in the General Case 207
15. CONTENTS xiii
5.7 Implications 211
5.8 Empirical Application: Calibrating a Real-Business-
Cycle Model 217
5.9 Empirical Application: Money and Output 220
5.10 Assessing the Baseline Real-Business-Cycle Model 226
Problems 233
Chapter 6 NOMINAL RIGIDITY 238
Part A EXOGENOUS NOMINAL RIGIDITY 239
6.1 A Baseline Case: Fixed Prices 239
6.2 Price Rigidity, Wage Rigidity, and Departures from
Perfect Competition in the Goods and Labor
Markets 244
6.3 Empirical Application: The Cyclical Behavior of the
Real Wage 253
6.4 Toward a Usable Model with Exogenous Nominal
Rigidity 255
Part B MICROECONOMIC FOUNDATIONS OF INCOMPLETE
NOMINAL ADJUSTMENT 267
6.5 A Model of Imperfect Competition and Price-Setting 268
6.6 Are Small Frictions Enough? 275
6.7 Real Rigidity 278
6.8 Coordination-Failure Models and Real Non-
Walrasian Theories 286
6.9 The Lucas Imperfect-Information Model 292
6.10 Empirical Application: International Evidence on the
Output-Inflation Tradeoff 302
Problems 306
Chapter 7 DYNAMIC STOCHASTIC GENERAL-
EQUILIBRIUM MODELS OF
FLUCTUATIONS 312
7.1 Building Blocks of Dynamic New Keynesian Models 315
7.2 Predetermined Prices: The Fischer Model 319
7.3 Fixed Prices: The Taylor Model 322
7.4 The Calvo Model and the New Keynesian Phillips
Curve 329
16. xiv CONTENTS
7.5 State-Dependent Pricing 332
7.6 Empirical Applications 337
7.7 Models of Staggered Price Adjustment with
Inflation Inertia 344
7.8 The Canonical New Keynesian Model 352
7.9 Other Elements of Modern New Keynesian DSGE
Models of Fluctuations 356
Problems 361
Chapter 8 CONSUMPTION 365
8.1 Consumption under Certainty: The Permanent-
Income Hypothesis 365
8.2 Consumption under Uncertainty: The Random-
Walk Hypothesis 372
8.3 Empirical Application: Two Tests of the Random-
Walk Hypothesis 375
8.4 The Interest Rate and Saving 380
8.5 Consumption and Risky Assets 384
8.6 Beyond the Permanent-Income Hypothesis 389
Problems 398
Chapter 9 INVESTMENT 405
9.1 Investment and the Cost of Capital 405
9.2 A Model of Investment with Adjustment Costs 408
9.3 Tobin’s q 414
9.4 Analyzing the Model 415
9.5 Implications 419
9.6 Empirical Application: q and Investment 425
9.7 The Effects of Uncertainty 428
9.8 Kinked and Fixed Adjustment Costs 432
9.9 Financial-Market Imperfections 436
9.10 Empirical Application: Cash Flow and Investment 447
Problems 451
Chapter 10 UNEMPLOYMENT 456
10.1 Introduction: Theories of Unemployment 456
10.2 A Generic Efficiency-Wage Model 458
10.3 A More General Version 463
17. CONTENTS xv
10.4 The Shapiro–Stiglitz Model 467
10.5 Contracting Models 478
10.6 Search and Matching Models 486
10.7 Implications 493
10.8 Empirical Applications 498
Problems 506
Chapter 11 INFLATION AND MONETARY
POLICY 513
11.1 Inflation, Money Growth, and Interest Rates 514
11.2 Monetary Policy and the Term Structure of
Interest Rates 518
11.3 The Microeconomic Foundations of Stabilization
Policy 523
11.4 Optimal Monetary Policy in a Simple Backward-
Looking Model 531
11.5 Optimal Monetary Policy in a Simple Forward-
Looking Model 537
11.6 Additional Issues in the Conduct of Monetary
Policy 542
11.7 The Dynamic Inconsistency of Low-Inflation
Monetary Policy 554
11.8 Empirical Applications 562
11.9 Seignorage and Inflation 567
Problems 576
Chapter 12 BUDGET DEFICITS AND FISCAL
POLICY 584
12.1 The Government Budget Constraint 586
12.2 The Ricardian Equivalence Result 592
12.3 Ricardian Equivalence in Practice 594
12.4 Tax-Smoothing 598
12.5 Political-Economy Theories of Budget Deficits 604
12.6 Strategic Debt Accumulation 607
12.7 Delayed Stabilization 617
12.8 Empirical Application: Politics and Deficits in
Industrialized Countries 623
12.9 The Costs of Deficits 628
12.10 A Model of Debt Crises 632
Problems 639
18. xvi CONTENTS
Epilogue THE FINANCIAL AND
MACROECONOMIC CRISIS OF 2008
AND BEYOND 644
References 649
Author Index 686
Subject Index 694
19. EMPIRICAL APPLICATIONS
Section 1.7 Growth Accounting 30
Convergence 32
Saving and Investment 36
Section 2.7 Wars and Real Interest Rates 75
Section 2.11 Are Modern Economies Dynamically Efficient? 90
Section 3.6 Time-Series Tests of Endogenous Growth Models 134
Section 3.7 Population Growth and Technological Change since
1 Million B.C. 138
Section 4.2 Accounting for Cross-Country Income Differences 156
Section 4.4 Social Infrastructure and Cross-Country Income
Differences 164
Section 4.5 Geography, Colonialism, and Economic Development 174
Section 5.8 Calibrating a Real-Business-Cycle Model 217
Section 5.9 Money and Output 220
Section 6.3 The Cyclical Behavior of the Real Wage 253
Section 6.8 Experimental Evidence on Coordination-Failure Games 289
Section 6.10 International Evidence on the Output-Inflation
Tradeoff 302
Section 7.6 Microeconomic Evidence on Price Adjustment 337
Inflation Inertia 340
Section 8.1 Understanding Estimated Consumption Functions 368
Section 8.3 Campbell and Mankiw’s Test Using Aggregate Data 375
Shea’s Test Using Household Data 377
Section 8.5 The Equity-Premium Puzzle 387
Section 8.6 Credit Limits and Borrowing 395
Section 9.6 q and Investment 425
Section 9.10 Cash Flow and Investment 447
Section 10.8 Contracting Effects on Employment 498
Interindustry Wage Differences 501
Survey Evidence on Wage Rigidity 504
Section 11.2 The Term Structure and Changes in the Federal
Reserve’s Funds-Rate Target 520
Section 11.6 Estimating Interest-Rate Rules 548
Section 11.8 Central-Bank Independence and Inflation 562
The Great Inflation 564
Section 12.1 Is U.S. Fiscal Policy on a Sustainable Path? 590
Section 12.8 Politics and Deficits in Industrialized Countries 623
xvii
21. PREFACE TO THE FOURTH
EDITION
Keeping a book on macroeconomics up to date is a challenging and never-
ending task. The field is continually evolving, as new events and research
lead to doubts about old views and the emergence of new ideas, models,
and tests. The result is that each edition of this book is very different from
the one before. This is truer of this revision than any previous one.
The largest changes are to the material on economic growth and on short-
run fluctuations with incomplete price flexibility. I have split the old chapter
on new growth theory in two. The first chapter (Chapter 3) covers models
of endogenous growth, and has been updated to include Paul Romer’s now-
classic model of endogenous technological progress. The second chapter
(Chapter 4) focuses on the enormous income differences across countries.
This material includes a much more extensive consideration of the chal-
lenges confronting empirical work on cross-country income differences and
of recent work on the underlying determinants of those differences.
Chapters 6 and 7 on short-run fluctuations when prices are not fully flex-
ible have been completely recast. This material is now grounded in micro-
economic foundations from the outset. It proceeds from simple models with
exogenously fixed prices to the microeconomic foundations of price sticki-
ness in static and dynamic settings, to the canonical three-equation new Key-
nesian model (the new Keynesian IS curve, the new Keynesian Phillips curve,
and an interest-rate rule), to the ingredients of modern dynamic stochastic
general-equilibrium models of fluctuations. These revisions carry over to the
analysis of monetary policy in Chapter 11. This chapter has been entirely
reorganized and is now much more closely tied to the earlier analyses of
short-run fluctuations, and it includes a careful treatment of optimal policy
in forward-looking models.
The two other chapters where I have made major changes are Chapter 5
on real-business-cycle models of fluctuations and Chapter 10 on the labor
market and unemployment. In Chapter 5, the empirical applications and the
analysis of the relation between real-business-cycle theory and other mod-
els of fluctuations have been overhauled. In Chapter 10, the presentation of
search-and-matching models of the labor market has been revamped and
greatly expanded, and the material on contracting models has been sub-
stantially compressed.
xix
22. xx PREFACE
Keeping the book up to date has been made even more challenging by
the financial and macroeconomic crisis that began in 2008. I have delib-
erately chosen not to change the book fundamentally in response to the
crisis: although I believe that the crisis will lead to major changes in macro-
economics, I also believe that it is too soon to know what those changes
will be. I have therefore taken the approach of bringing in the crisis where
it is relevant and of including an epilogue that describes some of the main
issues that the crisis raises for macroeconomics. But I believe that it will be
years before we have a clear picture of how the crisis is changing the field.
For additional reference and general information, please refer to the
book’s website at www.mhhe.com/romer4e. Also available on the website,
under the password-protected Instructor Edition, is the Solutions Manual.
Print versions of the manual are available by request only—if interested,
please contact your McGraw-Hill/Irwin representative.
This book owes a great deal to many people. The book is an outgrowth of
courses I have taught at Princeton University, the Massachusetts Institute of
Technology, Stanford University, and especially the University of California,
Berkeley. I want to thank the many students in these courses for their feed-
back, their patience, and their encouragement.
Four people have provided detailed, thoughtful, and constructive com-
ments on almost every aspect of the book over multiple editions: Laurence
Ball, A. Andrew John, N. Gregory Mankiw, and Christina Romer. Each has
significantly improved the book, and I am deeply grateful to them for their
efforts. In addition to those four, Susanto Basu, Robert Hall, and Ricardo
Reis provided extremely valuable guidance that helped shape the revisions
in this edition.
Many other people have made valuable comments and suggestions con-
cerning some or all of the book. I would particularly like to thank James
Butkiewicz, Robert Chirinko, Matthew Cushing, Charles Engel, Mark Gertler,
Robert Gordon, Mary Gregory, Tahereh Alavi Hojjat, A. Stephen Holland,
Hiroo Iwanari, Frederick Joutz, Pok-sang Lam, Gregory Linden, Maurice
Obtsfeld, Jeffrey Parker, Stephen Perez, Kerk Phillips, Carlos Ramirez,
Robert Rasche, Joseph Santos, Peter Skott, Peter Temin, Henry Thompson,
Matias Vernengo, and Steven Yamarik. Jeffrey Rohaly prepared the superb
Solutions Manual. Salifou Issoufou updated the tables and figures. Tyler
Arant, Zachary Breig, Chen Li, and Melina Mattos helped draft solutions
to the new problems and assisted with proofreading. Finally, the editorial
and production staff at McGraw-Hill did an excellent job of turning the
manuscript into a finished product. I thank all these people for their help.
23. INTRODUCTION
Macroeconomics is the study of the economy as a whole. It is therefore con-
cerned with some of the most important questions in economics. Why are
some countries rich and others poor? Why do countries grow? What are the
sources of recessions and booms? Why is there unemployment, and what
determines its extent? What are the sources of inflation? How do govern-
ment policies affect output, unemployment, inflation, and growth? These
and related questions are the subject of macroeconomics.
This book is an introduction to the study of macroeconomics at an ad-
vanced level. It presents the major theories concerning the central questions
of macroeconomics. Its goal is to provide both an overview of the field for
students who will not continue in macroeconomics and a starting point
for students who will go on to more advanced courses and research in
macroeconomics and monetary economics.
The book takes a broad view of the subject matter of macroeconomics. A
substantial portion of the book is devoted to economic growth, and separate
chapters are devoted to the natural rate of unemployment, inflation, and
budget deficits. Within each part, the major issues and competing theories
are presented and discussed. Throughout, the presentation is motivated
by substantive questions about the world. Models and techniques are used
extensively, but they are treated as tools for gaining insight into important
issues, not as ends in themselves.
The first four chapters are concerned with growth. The analysis focuses
on two fundamental questions: Why are some economies so much richer
than others, and what accounts for the huge increases in real incomes over
time? Chapter 1 is devoted to the Solow growth model, which is the basic
reference point for almost all analyses of growth. The Solow model takes
technological progress as given and investigates the effects of the division
of output between consumption and investment on capital accumulation
and growth. The chapter presents and analyzes the model and assesses its
ability to answer the central questions concerning growth.
Chapter 2 relaxes the Solow model’s assumption that the saving rate is
exogenous and fixed. It covers both a model where the set of households in
1
24. 2 INTRODUCTION
the economy is fixed (the Ramsey model) and one where there is turnover
(the Diamond model).
Chapter 3 presents the new growth theory. It begins with models where
technological progress arises from resources being devoted to the develop-
ment of new ideas, but where the division of resources between the produc-
tion of ideas and the production of conventional goods is taken as given. It
then considers the determinants of that division.
Chapter 4 focuses specifically on the sources of the enormous differ-
ences in average incomes across countries. This material, which is heavily
empirical, emphasizes two issues. The first is the contribution of variations
in the accumulation of physical and human capital and in output for given
quantities of capital to cross-country income differences. The other is the
determinants of those variations.
Chapters 5 through 7 are devoted to short-run fluctuations—the year-to-
year and quarter-to-quarter ups and downs of employment, unemployment,
and output. Chapter 5 investigates models of fluctuations where there are
no imperfections, externalities, or missing markets and where the economy
is subject only to real disturbances. This presentation of real-business-cycle
theory considers both a baseline model whose mechanics are fairly transpar-
ent and a more sophisticated model that incorporates additional important
features of fluctuations.
Chapters 6 and 7 then turn to Keynesian models of fluctuations. These
models are based on sluggish adjustment of nominal prices and wages,
and emphasize monetary as well as real disturbances. Chapter 6 focuses
on basic features of price stickiness. It investigates baseline models where
price stickiness is exogenous and the microeconomic foundations of price
stickiness in static settings. Chapter 7 turns to dynamics. It first exam-
ines the implications of alternative assumptions about price adjustment in
dynamic settings. It then turns to dynamic stochastic general-equilibrium
models of fluctuations with price stickiness—that is, fully specified general-
equilibrium models of fluctuations that incorporate incomplete nominal
price adjustment.
The analysis in the first seven chapters suggests that the behavior of
consumption and investment is central to both growth and fluctuations.
Chapters 8 and 9 therefore examine the determinants of consumption and
investment in more detail. In each case, the analysis begins with a baseline
model and then considers alternative views. For consumption, the baseline
is the permanent-income hypothesis; for investment, it is q theory.
Chapter 10 turns to the labor market. It focuses on the determinants of an
economy’s natural rate of unemployment. The chapter also investigates the
impact of fluctuations in labor demand on real wages and employment. The
main theories considered are efficiency-wage theories, contracting theories,
and search and matching models.
The final two chapters are devoted to macroeconomic policy. Chapter 11
investigates monetary policy and inflation. It starts by explaining the central
25. INTRODUCTION 3
role of money growth in causing inflation and by investigating the effects
of money growth. It then considers optimal monetary policy. This analysis
begins with the microeconomic foundations of the appropriate objective
for policy, proceeds to the analysis of optimal policy in backward-looking
and forward-looking models, and concludes with a discussion of a range of
issues in the conduct of policy. The final sections of the chapter examine
how excessive inflation can arise either from a short-run output-inflation
tradeoff or from governments’ need for revenue from money creation.
Chapter 12 is concerned with fiscal policy and budget deficits. The first
part of the chapter describes the government’s budget constraint and
investigates two baseline views of deficits: Ricardian equivalence and
tax-smoothing. Most of the remainder of the chapter investigates theories
of the sources of deficits. In doing so, it provides an introduction to the use
of economic tools to study politics.
Finally, a brief epilogue discusses the macroeconomic and financial crisis
that began in 2007 and worsened dramatically in the fall of 2008. The
focus is on the major issues that the crisis is likely to raise for the field
of macroeconomics.1
Macroeconomics is both a theoretical and an empirical subject. Because
of this, the presentation of the theories is supplemented with examples of
relevant empirical work. Even more so than with the theoretical sections, the
purpose of the empirical material is not to provide a survey of the literature;
nor is it to teach econometric techniques. Instead, the goal is to illustrate
some of the ways that macroeconomic theories can be applied and tested.
The presentation of this material is for the most part fairly intuitive and
presumes no more knowledge of econometrics than a general familiarity
with regressions. In a few places where it can be done naturally, the empir-
ical material includes discussions of the ideas underlying more advanced
econometric techniques.
Each chapter concludes with a set of problems. The problems range from
relatively straightforward variations on the ideas in the text to extensions
that tackle important issues. The problems thus serve both as a way for
readers to strengthen their understanding of the material and as a compact
way of presenting significant extensions of the ideas in the text.
The fact that the book is an advanced introduction to macroeconomics
has two main consequences. The first is that the book uses a series of for-
mal models to present and analyze the theories. Models identify particular
1
The chapters are largely independent. The growth and fluctuations sections are almost
entirely self-contained (although Chapter 5 builds moderately on Part A of Chapter 2). There
is also considerable independence among the chapters in each section. Chapters 2, 3, and 4
can be covered in any order, and models of price stickiness (Chapters 6 and 7) can be covered
either before or after real-business-cycle theory (Chapter 5). Finally, the last five chapters are
largely self-contained. The main exception is that Chapter 11 on monetary policy builds on
the analysis of models of fluctuations in Chapter 7. In addition, Chapter 8 relies moderately
on Chapter 2 and Chapter 10 relies moderately on Chapter 6.
26. 4 INTRODUCTION
features of reality and study their consequences in isolation. They thereby
allow us to see clearly how different elements of the economy interact and
what their implications are. As a result, they provide a rigorous way of
investigating whether a proposed theory can answer a particular question
and whether it generates additional predictions.
The book contains literally dozens of models. The main reason for this
multiplicity is that we are interested in many issues. Features of the econ-
omy that are crucial to one issue may be unimportant to others. Money, for
example, is almost surely central to inflation but not to long-run growth. In-
corporating money into models of growth would only obscure the analysis.
Thus instead of trying to build a single model to analyze all the issues we
are interested in, the book develops a series of models.
An additional reason for the multiplicity of models is that there is consid-
erable disagreement about the answers to many of the questions we will be
examining. When there is disagreement, the book presents the leading views
and discusses their strengths and weaknesses. Because different theories
emphasize different features of the economy, again it is more enlightening
to investigate distinct models than to build one model incorporating all the
features emphasized by the different views.
The second consequence of the book’s advanced level is that it presumes
some background in mathematics and economics. Mathematics provides
compact ways of expressing ideas and powerful tools for analyzing them.
The models are therefore mainly presented and analyzed mathematically.
The key mathematical requirements are a thorough understanding of single-
variable calculus and an introductory knowledge of multivariable calculus.
Tools such as functions, logarithms, derivatives and partial derivatives, max-
imization subject to constraint, and Taylor-series approximations are used
relatively freely. Knowledge of the basic ideas of probability—random vari-
ables, means, variances, covariances, and independence—is also assumed.
No mathematical background beyond this level is needed. More advanced
tools (such as simple differential equations, the calculus of variations, and
dynamic programming) are used sparingly, and they are explained as they
are used. Indeed, since mathematical techniques are essential to further
study and research in macroeconomics, models are sometimes analyzed in
greater detail than is otherwise needed in order to illustrate the use of a
particular method.
In terms of economics, the book assumes an understanding of microeco-
nomics through the intermediate level. Familiarity with such ideas as profit
maximization and utility maximization, supply and demand, equilibrium,
efficiency, and the welfare properties of competitive equilibria is presumed.
Little background in macroeconomics itself is absolutely necessary. Read-
ers with no prior exposure to macroeconomics, however, are likely to find
some of the concepts and terminology difficult, and to find that the pace is
rapid. These readers may wish to review an intermediate macroeconomics
27. INTRODUCTION 5
text before beginning the book, or to study such a book in conjunction with
this one.
The book was designed for first-year graduate courses in macroeco-
nomics. But it can be used (either on its own or in conjunction with an
intermediate text) for students with strong backgrounds in mathematics
and economics in professional schools and advanced undergraduate pro-
grams. It can also provide a tour of the field for economists and others
working in areas outside macroeconomics.
28. Chapter 1
THE SOLOW GROWTH MODEL
1.1 Some Basic Facts about Economic
Growth
Over the past few centuries, standards of living in industrialized countries
have reached levels almost unimaginable to our ancestors. Although com-
parisons are difficult, the best available evidence suggests that average real
incomes today in the United States and Western Europe are between 10 and
30 times larger than a century ago, and between 50 and 300 times larger
than two centuries ago.1
Moreover, worldwide growth is far from constant. Growth has been rising
over most of modern history. Average growth rates in the industrialized
countries were higher in the twentieth century than in the nineteenth, and
higher in the nineteenth than in the eighteenth. Further, average incomes
on the eve of the Industrial Revolution even in the wealthiest countries were
not dramatically above subsistence levels; this tells us that average growth
over the millennia before the Industrial Revolution must have been very,
very low.
One important exception to this general pattern of increasing growth
is the productivity growth slowdown. Average annual growth in output per
person in the United States and other industrialized countries from the early
1970s to the mid-1990s was about a percentage point below its earlier level.
The data since then suggest a rebound in productivity growth, at least in the
United States. How long the rebound will last and how widespread it will be
are not yet clear.
1
Maddison (2006) reports and discusses basic data on average real incomes over modern
history. Most of the uncertainty about the extent of long-term growth concerns the behav-
ior not of nominal income, but of the price indexes needed to convert those figures into
estimates of real income. Adjusting for quality changes and for the introduction of new
goods is conceptually and practically difficult, and conventional price indexes do not make
these adjustments well. See Nordhaus (1997) and Boskin, Dulberger, Gordon, Griliches, and
Jorgenson (1998) for discussions of the issues involved and analyses of the biases in con-
ventional price indexes.
6
29. 1.1 Some Basic Facts about Economic Growth 7
There are also enormous differences in standards of living across parts
of the world. Average real incomes in such countries as the United States,
Germany, and Japan appear to exceed those in such countries as Bangladesh
and Kenya by a factor of about 20.2
As with worldwide growth, cross-country
income differences are not immutable. Growth in individual countries often
differs considerably from average worldwide growth; that is, there are often
large changes in countries’ relative incomes.
The most striking examples of large changes in relative incomes are
growth miracles and growth disasters. Growth miracles are episodes where
growth in a country far exceeds the world average over an extended period,
with the result that the country moves rapidly up the world income distri-
bution. Some prominent growth miracles are Japan from the end of World
War II to around 1990, the newly industrializing countries (NICs) of East Asia
(South Korea, Taiwan, Singapore, and Hong Kong) starting around 1960, and
China starting around 1980. Average incomes in the NICs, for example, have
grown at an average annual rate of over 5 percent since 1960. As a result,
their average incomes relative to that of the United States have more than
tripled.
Growth disasters are episodes where a country’s growth falls far short
of the world average. Two very different examples of growth disasters are
Argentina and many of the countries of sub-Saharan Africa. In 1900,
Argentina’s average income was only slightly behind those of the world’s
leaders, and it appeared poised to become a major industrialized country.
But its growth performance since then has been dismal, and it is now near
the middle of the world income distribution. Sub-Saharan African countries
such as Chad, Ghana, and Mozambique have been extremely poor through-
out their histories and have been unable to obtain any sustained growth in
average incomes. As a result, their average incomes have remained close to
subsistence levels while average world income has been rising steadily.
Other countries exhibit more complicated growth patterns. Côte d’Ivoire
was held up as the growth model for Africa through the 1970s. From 1960 to
1978, real income per person grew at an average annual rate of 3.2 percent.
But in the three decades since then, its average income has not increased
at all, and it is now lower relative to that of the United States than it was in
1960. To take another example, average growth in Mexico was very high in
the 1950s, 1960s, and 1970s, negative in most of the 1980s, and moderate—
with a brief but severe interruption in the mid-1990s—since then.
Over the whole of the modern era, cross-country income differences have
widened on average. The fact that average incomes in the richest countries
at the beginning of the Industrial Revolution were not far above subsistence
2
Comparisons of real incomes across countries are far from straightforward, but are
much easier than comparisons over extended periods of time. The basic source for cross-
country data on real income is the Penn World Tables. Documentation of these data and the
most recent figures are available at http://pwt.econ.upenn.edu/.
30. 8 Chapter 1 THE SOLOW GROWTH MODEL
means that the overall dispersion of average incomes across different parts
of the world must have been much smaller than it is today (Pritchett, 1997).
Over the past few decades, however, there has been no strong tendency
either toward continued divergence or toward convergence.
The implications of the vast differences in standards of living over time
and across countries for human welfare are enormous. The differences are
associated with large differences in nutrition, literacy, infant mortality, life
expectancy, and other direct measures of well-being. And the welfare con-
sequences of long-run growth swamp any possible effects of the short-run
fluctuations that macroeconomics traditionally focuses on. During an av-
erage recession in the United States, for example, real income per person
falls by a few percent relative to its usual path. In contrast, the productivity
growth slowdown reduced real income per person in the United States by
about 25 percent relative to what it otherwise would have been. Other exam-
ples are even more startling. If real income per person in the Philippines con-
tinues to grow at its average rate for the period 1960–2001 of 1.5 percent, it
will take 150 years for it to reach the current U.S. level. If it achieves 3 per-
cent growth, the time will be reduced to 75 years. And if it achieves 5 percent
growth, as the NICs have done, the process will take only 45 years. To quote
Robert Lucas (1988), “Once one starts to think about [economic growth], it
is hard to think about anything else.”
The first four chapters of this book are therefore devoted to economic
growth. We will investigate several models of growth. Although we will
examine the models’ mechanics in considerable detail, our goal is to learn
what insights they offer concerning worldwide growth and income differ-
ences across countries. Indeed, the ultimate objective of research on eco-
nomic growth is to determine whether there are possibilities for raising
overall growth or bringing standards of living in poor countries closer to
those in the world leaders.
This chapter focuses on the model that economists have traditionally
used to study these issues, the Solow growth model.3
The Solow model is
the starting point for almost all analyses of growth. Even models that depart
fundamentally from Solow’s are often best understood through comparison
with the Solow model. Thus understanding the model is essential to under-
standing theories of growth.
The principal conclusion of the Solow model is that the accumulation
of physical capital cannot account for either the vast growth over time in
output per person or the vast geographic differences in output per per-
son. Specifically, suppose that capital accumulation affects output through
the conventional channel that capital makes a direct contribution to pro-
duction, for which it is paid its marginal product. Then the Solow model
3
The Solow model (which is sometimes known as the Solow–Swan model) was developed
by Robert Solow (Solow, 1956) and T. W. Swan (Swan, 1956).
31. 1.1 Some Basic Facts about Economic Growth 9
implies that the differences in real incomes that we are trying to under-
stand are far too large to be accounted for by differences in capital inputs.
The model treats other potential sources of differences in real incomes as
either exogenous and thus not explained by the model (in the case of tech-
nological progress, for example) or absent altogether (in the case of positive
externalities from capital, for example). Thus to address the central ques-
tions of growth theory, we must move beyond the Solow model.
Chapters 2 through 4 therefore extend and modify the Solow model.
Chapter 2 investigates the determinants of saving and investment. The
Solow model has no optimization in it; it takes the saving rate as exogenous
and constant. Chapter 2 presents two models that make saving endogenous
and potentially time-varying. In the first, saving and consumption decisions
are made by a fixed set of infinitely lived households; in the second, the
decisions are made by overlapping generations of households with finite
horizons.
Relaxing the Solow model’s assumption of a constant saving rate has
three advantages. First, and most important for studying growth, it demon-
strates that the Solow model’s conclusions about the central questions of
growth theory do not hinge on its assumption of a fixed saving rate. Second,
it allows us to consider welfare issues. A model that directly specifies rela-
tions among aggregate variables provides no way of judging whether some
outcomes are better or worse than others: without individuals in the model,
we cannot say whether different outcomes make individuals better or worse
off. The infinite-horizon and overlapping-generations models are built up
from the behavior of individuals, and can therefore be used to discuss wel-
fare issues. Third, infinite-horizon and overlapping-generations models are
used to study many issues in economics other than economic growth; thus
they are valuable tools.
Chapters 3 and 4 investigate more fundamental departures from the
Solow model. Their models, in contrast to Chapter 2’s, provide different
answers than the Solow model to the central questions of growth theory.
Chapter 3 departs from the Solow model’s treatment of technological pro-
gress as exogenous; it assumes instead that it is the result of the alloca-
tion of resources to the creation of new technologies. We will investigate
the implications of such endogenous technological progress for economic
growth and the determinants of the allocation of resources to innovative
activities.
The main conclusion of this analysis is that endogenous technological
progress is almost surely central to worldwide growth but probably has lit-
tle to do with cross-country income differences. Chapter 4 therefore focuses
specifically on those differences. We will find that understanding them re-
quires considering two new factors: variation in human as well as physical
capital, and variation in productivity not stemming from variation in tech-
nology. Chapter 4 explores both how those factors can help us understand
32. 10 Chapter 1 THE SOLOW GROWTH MODEL
the enormous differences in average incomes across countries and potential
sources of variation in those factors.
We now turn to the Solow model.
1.2 Assumptions
Inputs and Output
The Solow model focuses on four variables: output (Y ), capital (K ), labor
(L), and “knowledge” or the “effectiveness of labor” (A). At any time, the
economy has some amounts of capital, labor, and knowledge, and these are
combined to produce output. The production function takes the form
Y(t) = F (K(t),A(t)L(t)), (1.1)
where t denotes time.
Notice that time does not enter the production function directly, but only
through K, L, and A. That is, output changes over time only if the inputs
to production change. In particular, the amount of output obtained from
given quantities of capital and labor rises over time—there is technological
progress—only if the amount of knowledge increases.
Notice also that A and L enter multiplicatively. AL is referred to as effec-
tive labor, and technological progress that enters in this fashion is known as
labor-augmenting or Harrod-neutral.4
This way of specifying how A enters,
together with the other assumptions of the model, will imply that the ratio
of capital to output, K/Y, eventually settles down. In practice, capital-output
ratios do not show any clear upward or downward trend over extended peri-
ods. In addition, building the model so that the ratio is eventually constant
makes the analysis much simpler. Assuming that A multiplies L is therefore
very convenient.
The central assumptions of the Solow model concern the properties of the
production function and the evolution of the three inputs into production
(capital, labor, and knowledge) over time. We discuss each in turn.
Assumptions Concerning the Production Function
The model’s critical assumption concerning the production function is that
it has constant returns to scale in its two arguments, capital and effective
labor. That is, doubling the quantities of capital and effective labor (for ex-
ample, by doubling K and L with A held fixed) doubles the amount produced.
4
If knowledge enters in the form Y = F (AK,L), technological progress is capital-
augmenting. If it enters in the form Y = AF(K,L), technological progress is Hicks-neutral.
33. 1.2 Assumptions 11
More generally, multiplying both arguments by any nonnegative constant c
causes output to change by the same factor:
F (cK,cAL) = cF (K,AL) for all c ≥ 0. (1.2)
The assumption of constant returns can be thought of as a combination
of two separate assumptions. The first is that the economy is big enough that
the gains from specialization have been exhausted. In a very small economy,
there are likely to be enough possibilities for further specialization that
doubling the amounts of capital and labor more than doubles output. The
Solow model assumes, however, that the economy is sufficiently large that,
if capital and labor double, the new inputs are used in essentially the same
way as the existing inputs, and so output doubles.
The second assumption is that inputs other than capital, labor, and knowl-
edge are relatively unimportant. In particular, the model neglects land and
other natural resources. If natural resources are important, doubling capital
and labor could less than double output. In practice, however, as Section 1.8
describes, the availability of natural resources does not appear to be a major
constraint on growth. Assuming constant returns to capital and labor alone
therefore appears to be a reasonable approximation.
The assumption of constant returns allows us to work with the produc-
tion function in intensive form. Setting c = 1/AL in equation (1.2) yields
F
K
AL
,1
=
1
AL
F (K,AL). (1.3)
Here K/AL is the amount of capital per unit of effective labor, and F (K,AL)/
AL is Y/AL, output per unit of effective labor. Define k = K/AL, y = Y/AL,
and f (k) = F (k,1). Then we can rewrite (1.3) as
y = f (k). (1.4)
That is, we can write output per unit of effective labor as a function of
capital per unit of effective labor.
These new variables, k and y, are not of interest in their own right. Rather,
they are tools for learning about the variables we are interested in. As we
will see, the easiest way to analyze the model is to focus on the behavior
of k rather than to directly consider the behavior of the two arguments
of the production function, K and AL. For example, we will determine the
behavior of output per worker, Y/L, by writing it as A(Y/AL), or Af (k), and
determining the behavior of A and k.
To see the intuition behind (1.4), think of dividing the economy into AL
small economies, each with 1 unit of effective labor and K/AL units of capi-
tal. Since the production function has constant returns, each of these small
economies produces 1/AL as much as is produced in the large, undivided
economy. Thus the amount of output per unit of effective labor depends
only on the quantity of capital per unit of effective labor, and not on the over-
all size of the economy. This is expressed mathematically in equation (1.4).
34. 12 Chapter 1 THE SOLOW GROWTH MODEL
k
f(k)
FIGURE 1.1 An example of a production function
The intensive-form production function, f (k), is assumed to satisfy f (0) =
0, f ′
(k) 0, f ′′
(k) 0.5
Since F (K,AL) equals ALf (K/AL), it follows that
the marginal product of capital, ∂F (K,AL)/∂K, equals ALf ′
(K/AL)(1/AL),
which is just f ′
(k). Thus the assumptions that f ′
(k) is positive and f ′′
(k)
is negative imply that the marginal product of capital is positive, but that
it declines as capital (per unit of effective labor) rises. In addition, f (•)
is assumed to satisfy the Inada conditions (Inada, 1964): limk→0 f ′
(k) = ∞,
limk→∞ f ′
(k) = 0. These conditions (which are stronger than needed for the
model’s central results) state that the marginal product of capital is very
large when the capital stock is sufficiently small and that it becomes very
small as the capital stock becomes large; their role is to ensure that the path
of the economy does not diverge. A production function satisfying f ′
(•) 0,
f ′′
(•) 0, and the Inada conditions is shown in Figure 1.1.
A specific example of a production function is the Cobb–Douglas function,
F (K,AL) = Kα
(AL)1−α
, 0 α 1. (1.5)
This production function is easy to analyze, and it appears to be a good first
approximation to actual production functions. As a result, it is very useful.
5
The notation f ′
(•) denotes the first derivative of f (•), and f ′′
(•) the second derivative.
35. 1.2 Assumptions 13
It is easy to check that the Cobb–Douglas function has constant returns.
Multiplying both inputs by c gives us
F (cK,cAL) = (cK )α
(cAL)1−α
= cα
c1−α
Kα
(AL)1−α
= cF (K,AL).
(1.6)
To find the intensive form of the production function, divide both inputs
by AL; this yields
f (k) ≡ F
K
AL
,1
=
K
AL
α
= kα
.
(1.7)
Equation (1.7) implies that f ′
(k) = αk
α−1
. It is straightforward to check that
this expression is positive, that it approaches infinity as k approaches zero,
and that it approaches zero as k approaches infinity. Finally, f ′′
(k) =
−(1 − α)αk
α−2
, which is negative.6
The Evolution of the Inputs into Production
The remaining assumptions of the model concern how the stocks of labor,
knowledge, and capital change over time. The model is set in continuous
time; that is, the variables of the model are defined at every point in time.7
The initial levels of capital, labor, and knowledge are taken as given, and
are assumed to be strictly positive. Labor and knowledge grow at constant
rates:
L̇(t) = nL(t), (1.8)
Ȧ(t) = gA(t), (1.9)
where n and g are exogenous parameters and where a dot over a variable
denotes a derivative with respect to time (that is, Ẋ (t) is shorthand for
dX(t)/dt).
6
Note that with Cobb–Douglas production, labor-augmenting, capital-augmenting, and
Hicks-neutral technological progress (see n. 4) are all essentially the same. For example, to
rewrite (1.5) so that technological progress is Hicks-neutral, simply define à = A1−α
; then
Y = Ã(Kα
L1−α
).
7
The alternative is discrete time, where the variables are defined only at specific dates
(usually t = 0,1,2,. . .). The choice between continuous and discrete time is usually based on
convenience. For example, the Solow model has essentially the same implications in discrete
as in continuous time, but is easier to analyze in continuous time.
36. 14 Chapter 1 THE SOLOW GROWTH MODEL
The growth rate of a variable refers to its proportional rate of change.
That is, the growth rate of X refers to the quantity ˙
X (t)/X(t). Thus equa-
tion (1.8) implies that the growth rate of L is constant and equal to n, and
(1.9) implies that A’s growth rate is constant and equal to g.
A key fact about growth rates is that the growth rate of a variable equals
the rate of change of its natural log. That is, Ẋ (t)/X(t) equals d ln X(t)/dt. To
see this, note that since ln X is a function of X and X is a function of t, we
can use the chain rule to write
d ln X(t)
dt
=
d ln X(t)
dX(t)
dX(t)
dt
=
1
X(t)
Ẋ (t).
(1.10)
Applying the result that a variable’s growth rate equals the rate of change
of its log to (1.8) and (1.9) tells us that the rates of change of the logs of L
and A are constant and that they equal n and g, respectively. Thus,
ln L(t) = [ln L(0)] + nt, (1.11)
ln A(t) = [ln A(0)] + gt, (1.12)
where L(0) and A(0) are the values of L and A at time 0. Exponentiating both
sides of these equations gives us
L(t) = L(0)ent
, (1.13)
A(t) = A(0)egt
. (1.14)
Thus, our assumption is that L and A each grow exponentially.8
Output is divided between consumption and investment. The fraction
of output devoted to investment, s, is exogenous and constant. One unit of
output devoted to investment yields one unit of new capital. In addition,
existing capital depreciates at rate δ. Thus
K̇(t) = sY(t) − δK(t). (1.15)
Although no restrictions are placed on n, g, and δ individually, their sum is
assumed to be positive. This completes the description of the model.
Since this is the first model (of many!) we will encounter, this is a good
place for a general comment about modeling. The Solow model is grossly
simplified in a host of ways. To give just a few examples, there is only a
single good; government is absent; fluctuations in employment are ignored;
production is described by an aggregate production function with just three
inputs; and the rates of saving, depreciation, population growth, and tech-
nological progress are constant. It is natural to think of these features of
the model as defects: the model omits many obvious features of the world,
8
See Problems 1.1 and 1.2 for more on basic properties of growth rates.
37. 1.3 The Dynamics of the Model 15
and surely some of those features are important to growth. But the purpose
of a model is not to be realistic. After all, we already possess a model that
is completely realistic—the world itself. The problem with that “model” is
that it is too complicated to understand. A model’s purpose is to provide
insights about particular features of the world. If a simplifying assump-
tion causes a model to give incorrect answers to the questions it is being
used to address, then that lack of realism may be a defect. (Even then, the
simplification—by showing clearly the consequences of those features of
the world in an idealized setting—may be a useful reference point.) If the
simplification does not cause the model to provide incorrect answers to the
questions it is being used to address, however, then the lack of realism is
a virtue: by isolating the effect of interest more clearly, the simplification
makes it easier to understand.
1.3 The Dynamics of the Model
We want to determine the behavior of the economy we have just described.
The evolution of two of the three inputs into production, labor and knowl-
edge, is exogenous. Thus to characterize the behavior of the economy, we
must analyze the behavior of the third input, capital.
The Dynamics of k
Because the economy may be growing over time, it turns out to be much
easier to focus on the capital stock per unit of effective labor, k, than on the
unadjusted capital stock, K. Since k = K/AL, we can use the chain rule to
find
˙
k(t) =
K̇(t)
A(t)L(t)
−
K(t)
[A(t)L(t)]2
[A(t)L̇(t) + L(t)Ȧ(t)]
=
K̇(t)
A(t)L(t)
−
K(t)
A(t)L(t)
L̇(t)
L(t)
−
K(t)
A(t)L(t)
Ȧ(t)
A(t)
.
(1.16)
K/AL is simply k. From (1.8) and (1.9), L̇/L and Ȧ/A are n and g, respectively.
K̇ is given by (1.15). Substituting these facts into (1.16) yields
˙
k(t) =
sY(t) − δK(t)
A(t)L(t)
− k(t)n − k(t)g
= s
Y(t)
A(t)L(t)
− δk(t) − nk(t) − gk(t).
(1.17)
38. 16 Chapter 1 THE SOLOW GROWTH MODEL
k∗ k
sf(k)
Actual investment
Break-even investment
Investment
per
unit
of
effective
labor
(n + g + δ)k
FIGURE 1.2 Actual and break-even investment
Finally, using the fact that Y/AL is given by f (k), we have
˙
k(t) = sf (k(t)) − (n + g +δ)k(t). (1.18)
Equation (1.18) is the key equation of the Solow model. It states that
the rate of change of the capital stock per unit of effective labor is the
difference between two terms. The first, sf (k), is actual investment per unit
of effective labor: output per unit of effective labor is f (k), and the fraction
of that output that is invested is s. The second term, (n + g +δ)k, is break-
even investment, the amount of investment that must be done just to keep
k at its existing level. There are two reasons that some investment is needed
to prevent k from falling. First, existing capital is depreciating; this capital
must be replaced to keep the capital stock from falling. This is the δk term in
(1.18). Second, the quantity of effective labor is growing. Thus doing enough
investment to keep the capital stock (K ) constant is not enough to keep
the capital stock per unit of effective labor (k) constant. Instead, since the
quantity of effective labor is growing at rate n + g, the capital stock must
grow at rate n + g to hold k steady.9
This is the (n + g)k term in (1.18).
When actual investment per unit of effective labor exceeds the invest-
ment needed to break even, k is rising. When actual investment falls short
of break-even investment, k is falling. And when the two are equal, k is
constant.
Figure 1.2 plots the two terms of the expression for ˙
k as functions of k.
Break-even investment, (n + g+δ)k, is proportional to k. Actual investment,
sf (k), is a constant times output per unit of effective labor.
Since f (0) = 0, actual investment and break-even investment are equal at
k = 0. The Inada conditions imply that at k = 0, f ′
(k) is large, and thus that
the sf (k) line is steeper than the (n + g + δ)k line. Thus for small values of
9
The fact that the growth rate of the quantity of effective labor, AL, equals n + g is an
instance of the fact that the growth rate of the product of two variables equals the sum of
their growth rates. See Problem 1.1.
39. 1.3 The Dynamics of the Model 17
k
.
0
k
k∗
FIGURE 1.3 The phase diagram for k in the Solow model
k, actual investment is larger than break-even investment. The Inada con-
ditions also imply that f ′
(k) falls toward zero as k becomes large. At some
point, the slope of the actual investment line falls below the slope of the
break-even investment line. With the sf (k) line flatter than the (n + g + δ)k
line, the two must eventually cross. Finally, the fact that f ′′
(k) 0 implies
that the two lines intersect only once for k 0. We let k∗ denote the value
of k where actual investment and break-even investment are equal.
Figure 1.3 summarizes this information in the form of a phase diagram,
which shows ˙
k as a function of k. If k is initially less than k∗, actual in-
vestment exceeds break-even investment, and so ˙
k is positive—that is, k is
rising. If k exceeds k∗, ˙
k is negative. Finally, if k equals k∗, then ˙
k is zero.
Thus, regardless of where k starts, it converges to k∗ and remains there.10
The Balanced Growth Path
Since k converges to k∗, it is natural to ask how the variables of the model
behave when k equals k∗. By assumption, labor and knowledge are growing
at rates n and g, respectively. The capital stock, K, equals ALk; since k is
constant at k∗, K is growing at rate n + g (that is, K̇/K equals n + g). With
both capital and effective labor growing at rate n + g, the assumption of
constant returns implies that output, Y, is also growing at that rate. Finally,
capital per worker, K/L, and output per worker, Y/L, are growing at rate g.
10
If k is initially zero, it remains there. However, this possibility is ruled out by our
assumption that initial levels of K, L, and A are strictly positive.
40. 18 Chapter 1 THE SOLOW GROWTH MODEL
Thus the Solow model implies that, regardless of its starting point, the
economy converges to a balanced growth path—a situation where each
variable of the model is growing at a constant rate. On the balanced growth
path, the growth rate of output per worker is determined solely by the rate
of technological progress.11
1.4 The Impact of a Change in the
Saving Rate
The parameter of the Solow model that policy is most likely to affect is the
saving rate. The division of the government’s purchases between consump-
tion and investment goods, the division of its revenues between taxes and
borrowing, and its tax treatments of saving and investment are all likely to
affect the fraction of output that is invested. Thus it is natural to investigate
the effects of a change in the saving rate.
For concreteness, we will consider a Solow economy that is on a balanced
growth path, and suppose that there is a permanent increase in s. In addition
to demonstrating the model’s implications concerning the role of saving,
this experiment will illustrate the model’s properties when the economy is
not on a balanced growth path.
The Impact on Output
The increase in s shifts the actual investment line upward, and so k∗ rises.
This is shown in Figure 1.4. But k does not immediately jump to the new
value of k∗. Initially, k is equal to the old value of k∗. At this level, actual
investment now exceeds break-even investment—more resources are being
devoted to investment than are needed to hold k constant—and so ˙
k is
positive. Thus k begins to rise. It continues to rise until it reaches the new
value of k∗, at which point it remains constant.
These results are summarized in the first three panels of Figure 1.5. t0 de-
notes the time of the increase in the saving rate. By assumption, s jumps up
11
The broad behavior of the U.S. economy and many other major industrialized
economies over the last century or more is described reasonably well by the balanced growth
path of the Solow model. The growth rates of labor, capital, and output have each been
roughly constant. The growth rates of output and capital have been about equal (so that the
capital-output ratio has been approximately constant) and have been larger than the growth
rate of labor (so that output per worker and capital per worker have been rising). This is often
taken as evidence that it is reasonable to think of these economies as Solow-model economies
on their balanced growth paths. Jones (2002a) shows, however, that the underlying determi-
nants of the level of income on the balanced growth path have in fact been far from constant
in these economies, and thus that the resemblance between these economies and the bal-
anced growth path of the Solow model is misleading. We return to this issue in Section 3.3.
41. 1.4 The Impact of a Change in the Saving Rate 19
Investment
per
unit
of
effective
labor
k∗
OLD k∗
NEW
sNEWf(k)
k
(n + g + δ)k
sOLDf(k)
FIGURE 1.4 The effects of an increase in the saving rate on investment
at time t0 and remains constant thereafter. Since the jump in s causes actual
investment to exceed break-even investment by a strictly positive amount,
˙
k jumps from zero to a strictly positive amount. k rises gradually from the
old value of k∗ to the new value, and ˙
k falls gradually back to zero.12
We are likely to be particularly interested in the behavior of output per
worker, Y/L. Y/L equals Af (k). When k is constant, Y/L grows at rate g,
the growth rate of A. When k is increasing, Y/L grows both because A is
increasing and because k is increasing. Thus its growth rate exceeds g.
When k reaches the new value of k∗, however, again only the growth of
A contributes to the growth of Y/L, and so the growth rate of Y/L returns
to g. Thus a permanent increase in the saving rate produces a temporary
increase in the growth rate of output per worker: k is rising for a time, but
eventually it increases to the point where the additional saving is devoted
entirely to maintaining the higher level of k.
The fourth and fifth panels of Figure 1.5 show how output per worker
responds to the rise in the saving rate. The growth rate of output per worker,
which is initially g, jumps upward at t0 and then gradually returns to its
initial level. Thus output per worker begins to rise above the path it was on
and gradually settles into a higher path parallel to the first.13
12
For a sufficiently large rise in the saving rate, k̇ can rise for a while after t0 before
starting to fall back to zero.
13
Because the growth rate of a variable equals the derivative with respect to time of its
log, graphs in logs are often much easier to interpret than graphs in levels. For example, if
a variable’s growth rate is constant, the graph of its log as a function of time is a straight
line. This is why Figure 1.5 shows the log of output per worker rather than its level.
42. 20 Chapter 1 THE SOLOW GROWTH MODEL
s
k
0
c
t
t
t
t
t
t
t0
t0
t0
t0
t0
Growth
rate
of Y/L
ln(Y/L)
g
t0
k
.
FIGURE 1.5 The effects of an increase in the saving rate
In sum, a change in the saving rate has a level effect but not a growth
effect: it changes the economy’s balanced growth path, and thus the level of
output per worker at any point in time, but it does not affect the growth
rate of output per worker on the balanced growth path. Indeed, in the
43. 1.4 The Impact of a Change in the Saving Rate 21
Solow model only changes in the rate of technological progress have growth
effects; all other changes have only level effects.
The Impact on Consumption
If we were to introduce households into the model, their welfare would de-
pend not on output but on consumption: investment is simply an input into
production in the future. Thus for many purposes we are likely to be more
interested in the behavior of consumption than in the behavior of output.
Consumption per unit of effective labor equals output per unit of effec-
tive labor, f (k), times the fraction of that output that is consumed, 1 − s.
Thus, since s changes discontinuously at t0 and k does not, initially con-
sumption per unit of effective labor jumps downward. Consumption then
rises gradually as k rises and s remains at its higher level. This is shown in
the last panel of Figure 1.5.
Whether consumption eventually exceeds its level before the rise in s is
not immediately clear. Let c∗ denote consumption per unit of effective labor
on the balanced growth path. c∗ equals output per unit of effective labor,
f (k∗), minus investment per unit of effective labor, sf (k∗). On the balanced
growth path, actual investment equals break-even investment, (n + g+δ)k∗.
Thus,
c∗ = f (k∗) − (n + g +δ)k∗. (1.19)
k∗ is determined by s and the other parameters of the model, n, g, and δ;
we can therefore write k∗ = k∗(s,n,g,δ). Thus (1.19) implies
∂c∗
∂s
= [f ′
(k∗(s,n,g,δ)) − (n + g +δ)]
∂k∗(s,n,g,δ)
∂s
. (1.20)
We know that the increase in s raises k∗; that is, we know that ∂k∗/∂s
is positive. Thus whether the increase raises or lowers consumption in the
long run depends on whether f ′
(k∗)—the marginal product of capital—is
more or less than n + g+δ. Intuitively, when k rises, investment (per unit of
effective labor) must rise by n + g+δ times the change in k for the increase
to be sustained. If f ′
(k∗) is less than n + g + δ, then the additional output
from the increased capital is not enough to maintain the capital stock at
its higher level. In this case, consumption must fall to maintain the higher
capital stock. If f ′
(k∗) exceeds n + g + δ, on the other hand, there is more
than enough additional output to maintain k at its higher level, and so con-
sumption rises.
f ′
(k∗) can be either smaller or larger than n + g + δ. This is shown in
Figure 1.6. The figure shows not only (n + g+ δ)k and sf (k), but also f (k).
Since consumption on the balanced growth path equals output less break-
even investment (see [1.19]), c∗ is the distance between f (k) and (n + g+ δ)k
at k = k∗. The figure shows the determinants of c∗ for three different values
44. 22 Chapter 1 THE SOLOW GROWTH MODEL
Output
and
investment
per
unit
of
effective
labor
Output
and
investment
per
unit
of
effective
labor
Output
and
investment
per
unit
of
effective
labor
f (k)
sHf(k)
k
k
k
k∗
H
f (k)
f (k)
sMf(k)
k∗
L
k∗
M
(n + g + δ)k
(n + g + δ)k
(n + g + δ)k
sLf(k)
FIGURE 1.6 Output, investment, and consumption on the balanced growth
path
45. 1.5 Quantitative Implications 23
of s (and hence three different values of k∗). In the top panel, s is high, and
so k∗ is high and f ′
(k∗) is less than n + g + δ. As a result, an increase in
the saving rate lowers consumption even when the economy has reached
its new balanced growth path. In the middle panel, s is low, k∗ is low, f ′
(k∗)
is greater than n + g + δ, and an increase in s raises consumption in the
long run.
Finally, in the bottom panel, s is at the level that causes f ′
(k∗) to just equal
n + g+δ—that is, the f (k) and (n + g+δ)k loci are parallel at k = k∗. In this
case, a marginal change in s has no effect on consumption in the long run,
and consumption is at its maximum possible level among balanced growth
paths. This value of k∗ is known as the golden-rule level of the capital stock.
We will discuss the golden-rule capital stock further in Chapter 2. Among
the questions we will address are whether the golden-rule capital stock is
in fact desirable and whether there are situations in which a decentralized
economy with endogenous saving converges to that capital stock. Of course,
in the Solow model, where saving is exogenous, there is no more reason to
expect the capital stock on the balanced growth path to equal the golden-
rule level than there is to expect it to equal any other possible value.
1.5 Quantitative Implications
We are usually interested not just in a model’s qualitative implications, but
in its quantitative predictions. If, for example, the impact of a moderate
increase in saving on growth remains large after several centuries, the result
that the impact is temporary is of limited interest.
For most models, including this one, obtaining exact quantitative results
requires specifying functional forms and values of the parameters; it often
also requires analyzing the model numerically. But in many cases, it is possi-
ble to learn a great deal by considering approximations around the long-run
equilibrium. That is the approach we take here.
The Effect on Output in the Long Run
The long-run effect of a rise in saving on output is given by
∂y∗
∂s
= f ′
(k∗)
∂k∗(s,n,g,δ)
∂s
, (1.21)
where y∗ = f (k∗) is the level of output per unit of effective labor on the
balanced growth path. Thus to find ∂y∗/∂s, we need to find ∂k∗/∂s. To do
this, note that k∗ is defined by the condition that ˙
k = 0. Thus k∗ satisfies
sf (k∗(s,n,g,δ)) = (n + g +δ)k∗(s,n,g,δ). (1.22)
46. 24 Chapter 1 THE SOLOW GROWTH MODEL
Equation (1.22) holds for all values of s (and of n, g, and δ). Thus the deriva-
tives of the two sides with respect to s are equal:14
sf ′
(k∗)
∂k∗
∂s
+ f (k∗) = (n + g +δ)
∂k∗
∂s
, (1.23)
where the arguments of k∗ are omitted for simplicity. This can be rearranged
to obtain15
∂k∗
∂s
=
f (k∗)
(n + g +δ) − sf ′
(k∗)
. (1.24)
Substituting (1.24) into (1.21) yields
∂y∗
∂s
=
f ′
(k∗)f (k∗)
(n + g +δ) − sf ′
(k∗)
. (1.25)
Two changes help in interpreting this expression. The first is to convert it
to an elasticity by multiplying both sides by s/y∗. The second is to use the
fact that sf (k∗) = (n + g + δ)k∗ to substitute for s. Making these changes
gives us
s
y∗
∂y∗
∂s
=
s
f (k∗)
f ′
(k∗)f (k∗)
(n + g +δ) − sf ′
(k∗)
=
(n + g +δ)k∗f ′
(k∗)
f (k∗)[(n + g +δ) − (n + g +δ)k∗f ′
(k∗)/f (k∗)]
=
k∗f ′
(k∗)/f (k∗)
1 − [k∗f ′
(k∗)/f (k∗)]
.
(1.26)
k∗f ′
(k∗)/f (k∗) is the elasticity of output with respect to capital at k = k∗.
Denoting this by αK (k∗), we have
s
y∗
∂y∗
∂s
=
αK (k∗)
1 − αK (k∗)
. (1.27)
Thus we have found a relatively simple expression for the elasticity of the
balanced-growth-path level of output with respect to the saving rate.
To think about the quantitative implications of (1.27), note that if mar-
kets are competitive and there are no externalities, capital earns its marginal
14
This technique is known as implicit differentiation. Even though (1.22) does not ex-
plicitly give k∗ as a function of s, n, g, and δ, it still determines how k∗ depends on those
variables. We can therefore differentiate the equation with respect to s and solve for ∂k∗/∂s.
15
We saw in the previous section that an increase in s raises k∗. To check that this is
also implied by equation (1.24), note that n + g+ δ is the slope of the break-even investment
line and that sf ′
(k∗) is the slope of the actual investment line at k∗. Since the break-even
investment line is steeper than the actual investment line at k∗ (see Figure 1.2), it follows
that the denominator of (1.24) is positive, and thus that ∂k∗/∂s 0.
47. 1.5 Quantitative Implications 25
product. Since output equals ALf (k) and k equals K/AL, the marginal prod-
uct of capital, ∂Y/∂K, is ALf ′
(k)[1/(AL)], or just f ′
(k). Thus if capital earns its
marginal product, the total amount earned by capital (per unit of effective
labor) on the balanced growth path is k∗f ′
(k∗). The share of total income that
goes to capital on the balanced growth path is then k∗f ′
(k∗)/f (k∗), or αK (k∗).
In other words, if the assumption that capital earns its marginal product is
a good approximation, we can use data on the share of income going to
capital to estimate the elasticity of output with respect to capital, αK (k∗).
In most countries, the share of income paid to capital is about one-third.
If we use this as an estimate of αK (k∗), it follows that the elasticity of output
with respect to the saving rate in the long run is about one-half. Thus, for
example, a 10 percent increase in the saving rate (from 20 percent of output
to 22 percent, for instance) raises output per worker in the long run by about
5 percent relative to the path it would have followed. Even a 50 percent
increase in s raises y∗ only by about 22 percent. Thus significant changes
in saving have only moderate effects on the level of output on the balanced
growth path.
Intuitively, a small value of αK (k∗) makes the impact of saving on output
low for two reasons. First, it implies that the actual investment curve, sf (k),
bends fairly sharply. As a result, an upward shift of the curve moves its
intersection with the break-even investment line relatively little. Thus the
impact of a change in s on k∗ is small. Second, a low value of αK (k∗) means
that the impact of a change in k∗ on y∗ is small.
The Speed of Convergence
In practice, we are interested not only in the eventual effects of some change
(such as a change in the saving rate), but also in how rapidly those effects
occur. Again, we can use approximations around the long-run equilibrium
to address this issue.
For simplicity, we focus on the behavior of k rather than y. Our goal is thus
to determine how rapidly k approaches k∗. We know that ˙
k is determined
by k: recall that the key equation of the model is ˙
k = sf (k) − (n + g + δ)k
(see [1.18]). Thus we can write ˙
k = ˙
k(k). When k equals k∗, ˙
k is zero. A first-
order Taylor-series approximation of ˙
k(k) around k = k∗ therefore yields
˙
k ≃
∂˙
k(k)
∂k
k=k∗
(k − k∗). (1.28)
That is, ˙
k is approximately equal to the product of the difference between
k and k∗ and the derivative of ˙
k with respect to k at k = k∗.
Let λ denote −∂˙
k(k)/∂k|k=k∗ . With this definition, (1.28) becomes
˙
k(t) ≃ −λ[k(t) − k∗]. (1.29)
48. 26 Chapter 1 THE SOLOW GROWTH MODEL
Since ˙
k is positive when k is slightly below k∗ and negative when it is slightly
above, ∂˙
k(k)/∂k|k=k∗ is negative. Equivalently, λ is positive.
Equation (1.29) implies that in the vicinity of the balanced growth path,
k moves toward k∗ at a speed approximately proportional to its distance
from k∗. That is, the growth rate of k(t) − k∗ is approximately constant and
equal to −λ. This implies
k(t) ≃ k∗ + e−λt
[k(0) − k∗], (1.30)
where k(0) is the initial value of k. Note that (1.30) follows just from the
facts that the system is stable (that is, that k converges to k∗) and that we
are linearizing the equation for ˙
k around k = k∗.
It remains to find λ; this is where the specifics of the model enter the anal-
ysis. Differentiating expression (1.18) for ˙
k with respect to k and evaluating
the resulting expression at k = k∗ yields
λ ≡ −
∂˙
k(k)
∂k
k=k∗
= −[sf ′
(k∗) − (n + g +δ)]
= (n + g +δ) − sf ′
(k∗)
= (n + g +δ) −
(n + g + δ)k∗f ′
(k∗)
f (k∗)
= [1 − αK (k∗)](n + g + δ).
(1.31)
Here the third line again uses the fact that sf (k∗) = (n + g + δ)k∗ to sub-
stitute for s, and the last line uses the definition of αK . Thus, k converges
to its balanced-growth-path value at rate [1 − αK (k∗)](n + g+δ). In addition,
one can show that y approaches y∗ at the same rate that k approaches k∗.
That is, y(t) − y∗ ≃ e−λt
[y(0) − y∗].16
We can calibrate (1.31) to see how quickly actual economies are likely to
approach their balanced growth paths. Typically, n+g+δis about 6 percent
per year. This arises, for example, with 1 to 2 percent population growth, 1
to 2 percent growth in output per worker, and 3 to 4 percent depreciation.
If capital’s share is roughly one-third, (1 − αK )(n + g + δ) is thus roughly
4 percent. Therefore k and y move 4 percent of the remaining distance
toward k∗ and y∗ each year, and take approximately 17 years to get halfway
to their balanced-growth-path values.17
Thus in our example of a 10 percent
16
See Problem 1.11.
17
The time it takes for a variable (in this case, y − y∗) with a constant negative growth rate
to fall in half is approximately equal to 70 divided by its growth rate in percent. (Similarly,
the doubling time of a variable with positive growth is 70 divided by the growth rate.) Thus
in this case the half-life is roughly 70/(4%/year), or about 17 years. More exactly, the half-life,
t∗, is the solution to e−λt ∗
= 0.5, where λ is the rate of decrease. Taking logs of both sides,
t∗ = − ln(0.5)/λ ≃ 0.69/λ.
49. 1.6 The Solow Model and the Central Questions of Growth Theory 27
increase in the saving rate, output is 0.04(5%) = 0.2% above its previous
path after 1 year; is 0.5(5%) = 2.5% above after 17 years; and asymptotically
approaches 5 percent above the previous path. Thus not only is the overall
impact of a substantial change in the saving rate modest, but it does not
occur very quickly.18
1.6 The Solow Model and the Central
Questions of Growth Theory
The Solow model identifies two possible sources of variation—either over
time or across parts of the world—in output per worker: differences in cap-
ital per worker (K/L) and differences in the effectiveness of labor (A). We
have seen, however, that only growth in the effectiveness of labor can lead
to permanent growth in output per worker, and that for reasonable cases
the impact of changes in capital per worker on output per worker is modest.
As a result, only differences in the effectiveness of labor have any reason-
able hope of accounting for the vast differences in wealth across time and
space. Specifically, the central conclusion of the Solow model is that if the
returns that capital commands in the market are a rough guide to its con-
tributions to output, then variations in the accumulation of physical capital
do not account for a significant part of either worldwide economic growth
or cross-country income differences.
There are two ways to see that the Solow model implies that differ-
ences in capital accumulation cannot account for large differences in in-
comes, one direct and the other indirect. The direct approach is to con-
sider the required differences in capital per worker. Suppose we want to
account for a difference of a factor of X in output per worker between
two economies on the basis of differences in capital per worker. If out-
put per worker differs by a factor of X, the difference in log output per
worker between the two economies is ln X. Since the elasticity of output per
worker with respect to capital per worker is αK , log capital per worker must
differ by (ln X )/αK . That is, capital per worker differs by a factor of e(ln X )/αK
,
or X 1/αK
.
Output per worker in the major industrialized countries today is on the
order of 10 times larger than it was 100 years ago, and 10 times larger than
it is in poor countries today. Thus we would like to account for values of
18
These results are derived from a Taylor-series approximation around the balanced
growth path. Thus, formally, we can rely on them only in an arbitrarily small neighborhood
around the balanced growth path. The question of whether Taylor-series approximations
provide good guides for finite changes does not have a general answer. For the Solow model
with conventional production functions, and for moderate changes in parameter values (such
as those we have been considering), the Taylor-series approximations are generally quite
reliable.
50. 28 Chapter 1 THE SOLOW GROWTH MODEL
X in the vicinity of 10. Our analysis implies that doing this on the basis of
differences in capital requires a difference of a factor of 101/αK
in capital
per worker. For αK = 1
3
, this is a factor of 1000. Even if capital’s share is
one-half, which is well above what data on capital income suggest, one still
needs a difference of a factor of 100.
There is no evidence of such differences in capital stocks. Capital-output
ratios are roughly constant over time. Thus the capital stock per worker in
industrialized countries is roughly 10 times larger than it was 100 years
ago, not 100 or 1000 times larger. Similarly, although capital-output ratios
vary somewhat across countries, the variation is not great. For example,
the capital-output ratio appears to be 2 to 3 times larger in industrialized
countries than in poor countries; thus capital per worker is “only” about 20
to 30 times larger. In sum, differences in capital per worker are far smaller
than those needed to account for the differences in output per worker that
we are trying to understand.
The indirect way of seeing that the model cannot account for large varia-
tions in output per worker on the basis of differences in capital per worker is
to notice that the required differences in capital imply enormous differences
in the rate of return on capital (Lucas, 1990). If markets are competitive, the
rate of return on capital equals its marginal product, f ′
(k), minus depreci-
ation, δ. Suppose that the production function is Cobb–Douglas, which in
intensive form is f (k) = kα
(see equation [1.7]). With this production func-
tion, the elasticity of output with respect to capital is simply α. The marginal
product of capital is
f ′
(k) = αkα−1
= αy (α−1)/α
.
(1.32)
Equation (1.32) implies that the elasticity of the marginal product of cap-
ital with respect to output is −(1 − α)/α. If α= 1
3
, a tenfold difference in
output per worker arising from differences in capital per worker thus im-
plies a hundredfold difference in the marginal product of capital. And since
the return to capital is f ′
(k) − δ, the difference in rates of return is even
larger.
Again, there is no evidence of such differences in rates of return. Direct
measurement of returns on financial assets, for example, suggests only
moderate variation over time and across countries. More tellingly, we can
learn much about cross-country differences simply by examining where the
holders of capital want to invest. If rates of return were larger by a factor of
10 or 100 in poor countries than in rich countries, there would be immense
incentives to invest in poor countries. Such differences in rates of return
would swamp such considerations as capital-market imperfections, govern-
ment tax policies, fear of expropriation, and so on, and we would observe
51. 1.6 The Solow Model and the Central Questions of Growth Theory 29
immense flows of capital from rich to poor countries. We do not see such
flows.19
Thus differences in physical capital per worker cannot account for the
differences in output per worker that we observe, at least if capital’s con-
tribution to output is roughly reflected by its private returns.
The other potential source of variation in output per worker in the Solow
model is the effectiveness of labor. Attributing differences in standards of
living to differences in the effectiveness of labor does not require huge dif-
ferences in capital or in rates of return. Along a balanced growth path, for
example, capital is growing at the same rate as output; and the marginal
product of capital, f ′
(k), is constant.
Unfortunately, however, the Solow model has little to say about the effec-
tiveness of labor. Most obviously, the growth of the effectiveness of labor
is exogenous: the model takes as given the behavior of the variable that it
identifies as the driving force of growth. Thus it is only a small exaggeration
to say that we have been modeling growth by assuming it.
More fundamentally, the model does not identify what the “effectiveness
of labor” is; it is just a catchall for factors other than labor and capital
that affect output. Thus saying that differences in income are due to dif-
ferences in the effectiveness of labor is no different than saying that they
are not due to differences in capital per worker. To proceed, we must take
a stand concerning what we mean by the effectiveness of labor and what
causes it to vary. One natural possibility is that the effectiveness of labor
corresponds to abstract knowledge. To understand worldwide growth, it
would then be necessary to analyze the determinants of the stock of knowl-
edge over time. To understand cross-country differences in real incomes,
one would have to explain why firms in some countries have access to more
knowledge than firms in other countries, and why that greater knowledge is
not rapidly transmitted to poorer countries.
There are other possible interpretations of A: the education and skills of
the labor force, the strength of property rights, the quality of infrastructure,
cultural attitudes toward entrepreneurship and work, and so on. Or A may
reflect a combination of forces. For any proposed view of what A represents,
one would again have to address the questions of how it affects output, how
it evolves over time, and why it differs across parts of the world.
The other possible way to proceed is to consider the possibility that capi-
tal is more important than the Solow model implies. If capital encompasses
19
One can try to avoid this conclusion by considering production functions where capi-
tal’s marginal product falls less rapidly ask rises than it does in the Cobb–Douglas case. This
approach encounters two major difficulties. First, since it implies that the marginal product
of capital is similar in rich and poor countries, it implies that capital’s share is much larger
in rich countries. Second, and similarly, it implies that real wages are only slightly larger in
rich than in poor countries. These implications appear grossly inconsistent with the facts.
52. 30 Chapter 1 THE SOLOW GROWTH MODEL
more than just physical capital, or if physical capital has positive external-
ities, then the private return on physical capital is not an accurate guide to
capital’s importance in production. In this case, the calculations we have
done may be misleading, and it may be possible to resuscitate the view that
differences in capital are central to differences in incomes.
These possibilities for addressing the fundamental questions of growth
theory are the subject of Chapters 3 and 4.
1.7 Empirical Applications
Growth Accounting
In many situations, we are interested in the proximate determinants of
growth. That is, we often want to know how much of growth over some
period is due to increases in various factors of production, and how much
stems from other forces. Growth accounting, which was pioneered by
Abramovitz (1956) and Solow (1957), provides a way of tackling this subject.
To see how growth accounting works, consider again the production func-
tion Y(t) = F (K(t),A(t)L(t)). This implies
Ẏ (t) =
∂Y(t)
∂K(t)
K̇(t) +
∂Y(t)
∂L(t)
L̇(t) +
∂Y(t)
∂A(t)
Ȧ(t), (1.33)
where ∂Y/∂L and ∂Y/∂A denote [∂Y/∂(AL)]A and [∂Y/∂(AL)]L, respectively.
Dividing both sides by Y(t) and rewriting the terms on the right-hand side
yields
Ẏ (t)
Y(t)
=
K(t)
Y(t)
∂Y(t)
∂K(t)
K̇(t)
K(t)
+
L(t)
Y(t)
∂Y(t)
∂L(t)
L̇(t)
L(t)
+
A(t)
Y(t)
∂Y(t)
∂A(t)
Ȧ(t)
A(t)
≡ αK (t)
K̇(t)
K(t)
+ αL(t)
L̇(t)
L(t)
+ R(t).
(1.34)
Here αL(t) is the elasticity of output with respect to labor at time t,
αK (t) is again the elasticity of output with respect to capital, and R(t) ≡
[A(t)/Y(t)][∂Y(t)/∂A(t)][Ȧ(t)/A(t)]. Subtracting L̇(t)/L(t) from both sides and
using the fact that αL(t) + αK (t) = 1 (see Problem 1.9) gives an expression
for the growth rate of output per worker:
Ẏ (t)
Y(t)
−
L̇(t)
L(t)
= αK (t)
K̇(t)
K(t)
−
L̇(t)
L(t)
+ R(t). (1.35)
The growth rates of Y, K, and L are straightforward to measure. And we
know that if capital earns its marginal product, αK can be measured using
data on the share of income that goes to capital. R(t) can then be mea-
sured as the residual in (1.35). Thus (1.35) provides a way of decomposing
the growth of output per worker into the contribution of growth of capital
per worker and a remaining term, the Solow residual. The Solow residual
53. 1.7 Empirical Applications 31
is sometimes interpreted as a measure of the contribution of technological
progress. As the derivation shows, however, it reflects all sources of growth
other than the contribution of capital accumulation via its private return.
This basic framework can be extended in many ways. The most common
extensions are to consider different types of capital and labor and to adjust
for changes in the quality of inputs. But more complicated adjustments are
also possible. For example, if there is evidence of imperfect competition,
one can try to adjust the data on income shares to obtain a better estimate
of the elasticity of output with respect to the different inputs.
Growth accounting only examines the immediate determinants of growth:
it asks how much factor accumulation, improvements in the quality of in-
puts, and so on contribute to growth while ignoring the deeper issue of
what causes the changes in those determinants. One way to see that growth
accounting does not get at the underlying sources of growth is to consider
what happens if it is applied to an economy described by the Solow model
that is on its balanced growth path. We know that in this case growth is com-
ing entirely from growth in A. But, as Problem 1.13 asks you to show and
explain, growth accounting in this case attributes only fraction 1 − αK (k∗)
of growth to the residual, and fraction αK (k∗) to capital accumulation.
Even though growth accounting provides evidence only about the im-
mediate sources of growth, it has been fruitfully applied to many issues.
For example, it has played a major role in a recent debate concerning the
exceptionally rapid growth of the newly industrializing countries of East
Asia. Young (1995) uses detailed growth accounting to argue that the higher
growth in these countries than in the rest of the world is almost entirely due
to rising investment, increasing labor force participation, and improving
labor quality (in terms of education), and not to rapid technological progress
and other forces affecting the Solow residual. This suggests that for other
countries to replicate the NICs’ successes, it is enough for them to promote
accumulation of physical and human capital and greater use of resources,
and that they need not tackle the even more difficult task of finding ways
of obtaining greater output for a given set of inputs. In this view, the NICs’
policies concerning trade, regulation, and so on have been important largely
only to the extent they have influenced factor accumulation and factor use.
Hsieh (2002), however, observes that one can do growth accounting by
examining the behavior of factor returns rather than quantities. If rapid
growth comes solely from capital accumulation, for example, we will see
either a large fall in the return to capital or a large rise in capital’s share
(or a combination). Doing the growth accounting this way, Hsieh finds a
much larger role for the residual. Young (1998) and Fernald and Neiman
(2008) extend the analysis further, and identify reasons that Hsieh’s analysis
may have underestimated the role of factor accumulation.
Growth accounting has also been used extensively to study both the pro-
ductivity growth slowdown (the reduced growth rate of output per worker-
hour in the United States and other industrialized countries that began
54. 32 Chapter 1 THE SOLOW GROWTH MODEL
in the early 1970s) and the productivity growth rebound (the return of U.S.
productivity growth starting in the mid-1990s to close to its level before the
slowdown). Growth-accounting studies of the rebound suggest that comput-
ers and other types of information technology are the main source of the
rebound (see, for example, Oliner and Sichel, 2002, and Oliner, Sichel, and
Stiroh, 2007). Until the mid-1990s, the rapid technological progress in com-
puters and their introduction in many sectors of the economy appear to
have had little impact on aggregate productivity. In part, this was simply
because computers, although spreading rapidly, were still only a small frac-
tion of the overall capital stock. And in part, it was because the adoption
of the new technologies involved substantial adjustment costs. The growth-
accounting studies find, however, that since the mid-1990s, computers and
other forms of information technology have had a large impact on aggregate
productivity.20
Convergence
An issue that has attracted considerable attention in empirical work on
growth is whether poor countries tend to grow faster than rich countries.
There are at least three reasons that one might expect such convergence.
First, the Solow model predicts that countries converge to their balanced
growth paths. Thus to the extent that differences in output per worker arise
from countries being at different points relative to their balanced growth
paths, one would expect poor countries to catch up to rich ones. Second, the
Solow model implies that the rate of return on capital is lower in countries
with more capital per worker. Thus there are incentives for capital to flow
from rich to poor countries; this will also tend to cause convergence. And
third, if there are lags in the diffusion of knowledge, income differences
can arise because some countries are not yet employing the best available
technologies. These differences might tend to shrink as poorer countries
gain access to state-of-the-art methods.
Baumol (1986) examines convergence from 1870 to 1979 among the 16
industrialized countries for which Maddison (1982) provides data. Baumol
regresses output growth over this period on a constant and initial income.
20
The simple information-technology explanation of the productivity growth rebound
faces an important challenge, however: other industrialized countries have for the most part
not shared in the rebound. The leading candidate explanation of this puzzle is closely related
to the observation that there are large adjustments costs in adopting the new technologies.
In this view, the adoption of computers and information technology raises productivity
substantially only if it is accompanied by major changes in worker training, the composition
of the firm’s workforce, and the organization of the firm. Thus in countries where firms
lack the ability to make these changes (because of either government regulation or business
culture), the information-technology revolution is, as yet, having little impact on overall
economic performance (see, for example, Breshnahan, Brynjolfsson, and Hitt, 2002; Basu,
Fernald, Oulton, and Srinivasan, 2003; and Bloom, Sadun, and Van Reenan, 2008).
55. 1.7 Empirical Applications 33
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Japan
Sweden
Finland
Norway
Germany
Austria
Italy
Canada
United States
Denmark
Switzerland
+ Belgium
Netherlands
United Kingdom
Australia
France
Log per capita income in 1870
Log
per
capita
income
growth
1870–1979
FIGURE 1.7 Initial income and subsequent growth in Baumol’s sample (from
DeLong, 1988; used with permission)
That is, he estimates
ln
Y
N
i,1979
− ln
Y
N
i,1870
= a + b ln
Y
N
i,1870
+ εi . (1.36)
Here ln(Y/N) is log income per person, ε is an error term, and i indexes coun-
tries.21
If there is convergence, b will be negative: countries with higher ini-
tial incomes have lower growth. A value for b of −1 corresponds to perfect
convergence: higher initial income on average lowers subsequent growth
one-for-one, and so output per person in 1979 is uncorrelated with its value
in 1870. A value for b of 0, on the other hand, implies that growth is uncor-
related with initial income and thus that there is no convergence.
The results are
ln
Y
N
i,1979
− ln
Y
N
i,1870
= 8.457 − 0.995
(0.094)
ln
Y
N
i,1870
,
(1.37)
R2
= 0.87, s.e.e. = 0.15,
where the number in parentheses, 0.094, is the standard error of the re-
gression coefficient. Figure 1.7 shows the scatterplot corresponding to this
regression.
The regression suggests almost perfect convergence. The estimate of b
is almost exactly equal to −1, and it is estimated fairly precisely; the
21
Baumol considers output per worker rather than output per person. This choice has
little effect on the results.